2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208140In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{xx}+iλ\abs{u}^k u_x=0$ with non-zero $λ\in \Real$ and any real number $k\gs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.25 pagesAnalysis of PDEs35Q55; 35A07text